Concept
In algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression. Examples include which arises in discussing the regular pentagon, and more complicated ones such as Some nested radicals can be rewritten in a form that is not nested. For example, Another simple example, Rewriting a nested radical in this way is called denesting. This is not always possible, and, even when possible, it is often difficult. In the case of two nested square roots, the following theorem completely solves the problem of denesting. If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that if and only if is the square of a rational number d. If the nested radical is real, x and y are the two numbers and where is a rational number. In particular, if a and c are integers, then 2x and 2y are integers. This result includes denestings of the form as z may always be written and at least one of the terms must be positive (because the left-hand side of the equation is positive). A more general denesting formula could have the form However, Galois theory implies that either the left-hand side belongs to or it must be obtained by changing the sign of either or both. In the first case, this means that one can take x = c and In the second case, and another coefficient must be zero. If one may rename xy as x for getting Proceeding similarly if it results that one can suppose This shows that the apparently more general denesting can always be reduced to the above one. Proof: By squaring, the equation is equivalent with and, in the case of a minus in the right-hand side, (square roots are nonnegative by definition of the notation). As the inequality may always be satisfied by possibly exchanging x and y, solving the first equation in x and y is equivalent with solving This equality implies that belongs to the quadratic field In this field every element may be uniquely written with and being rational numbers.